We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to 0, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origen. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.

中文翻译：

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具有多项式符号的 S(R) 上的复合算子的频谱

我们研究了快速递减函数的 Schwartz 空间中的算子谱，这些函数将每个函数与其组成与多项式联系起来。在这个算子是平均遍历的情况下，我们证明它的频谱减少到 0，而任何具有多项式的非平均遍历复合算子的频谱总是包含封闭的单位圆盘，除了原点。我们用二次多项式或带正系数的三次多项式得到了复合算子的频谱的完整描述。